On hereditarily self-similar -adic analytic pro- groups
Francesco Noseda
Federal University of Rio de Janeiro, BrazilIlir Snopce
Federal University of Rio de Janeiro, Brazil
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Abstract
A non-trivial finitely generated pro- group is said to be strongly hereditarily self-similar of index if every non-trivial finitely generated closed subgroup of admits a faithful self-similar action on a -ary tree. We classify the solvable torsion-free -adic analytic pro- groups of dimension less than that are strongly hereditarily self-similar of index . Moreover, we show that a solvable torsion-free -adic analytic pro- group of dimension less than is strongly hereditarily self-similar of index if and only if it is isomorphic to the maximal pro- Galois group of some field that contains a primitive th root of unity. As a key step for the proof of the above results, we classify the -dimensional solvable torsion-free -adic analytic pro- groups that admit a faithful self-similar action on a -ary tree, completing the classification of the -dimensional torsion-free -adic analytic pro- groups that admit such actions.
Cite this article
Francesco Noseda, Ilir Snopce, On hereditarily self-similar -adic analytic pro- groups. Groups Geom. Dyn. 16 (2022), no. 1, pp. 85–114
DOI 10.4171/GGD/641